"In middle school, explanations should include mathematical arguments and rationales, not just procedural descriptions or summaries"

(NCTM, 2000, p. 268).

Let's begin our investigation by looking at a problem that is embedded in a game called Fraction Tracks, which will provide background for understanding the rest of this section. As you observe the game, think about how you might develop strategies to complete it and how you would communicate your thinking to another person.

The Fraction Tracks game can be played either cooperatively or competitively by two or more players. The game begins with a marker at 0 on each track. The object of the game is to get all the markers to 1. Players draw cards, on which appear all fractions greater than 0 and less than 1 with denominators of 2, 3, 4, 5, 6, 8, and 10, and they move one or more of their markers the total distance indicated on the card. Each player may move along different tracks in the same turn as long as the moves add up to the exact amount on the card. Since players may not "run over" 1 -- they must reach 1 by an exact amount -- the fraction a player draws might be too large, and that player might not be able to move without going over. If so, that player loses his or her turn. It can take several turns for a player to reach 1, depending on the cards drawn.

The goal of the game is to encourage mathematical discourse and to enhance the players' understanding of fractions.

We begin the game already in progress. Although Fraction Tracks is typically played with two people, for purposes of introducing you to the thinking and communication involved in the game, we would like you to observe one round of play and its possible outcomes. Note that this is not a display of the complete game. If you would like to play Fraction Tracks in its entirety, go to the NCTM Web site.